Non-Markovian stochastic processes and their applications: from anomalous diffusion to time series analysis

This work provides a forward step in the study and comprehension of the relationships between stochastic processes and a certain class of integral-partial differential equation, which can be used in order to model anomalous diffusion and transport in statistical physics. In the first part, we brought the reader through the fundamental notions of probability and stochastic processes, stochastic integration and stochastic differential equations as well. In particular, within the study of H-sssi processes, we focused on fractional Brownian motion (fBm) and its discrete-time increment process, the fractional Gaussian noise (fGn), which provide examples of non-Markovian Gaussian processes. The fGn, together with stationary FARIMA processes, is widely used in the modeling and estimation of long-memory, or long-range dependence (LRD). Time series manifesting long-range dependence, are often observed in nature especially in physics, meteorology, climatology, but also in hydrology, geophysics, economy and many others. We deepely studied LRD, giving many real data examples, providing statistical analysis and introducing parametric methods of estimation. Then, we introduced the theory of fractional integrals and derivatives, which indeed turns out to be very appropriate for studying and modeling systems with long-memory properties. After having introduced the basics concepts, we provided many examples and applications. For instance, we investigated the relaxation equation with distributed order time-fractional derivatives, which describes models characterized by a strong memory component and can be used to model relaxation in complex systems, which deviates from the classical exponential Debye pattern. Then, we focused in the study of generalizations of the standard diffusion equation, by passing through the preliminary study of the fractional forward drift equation. Such generalizations have been obtained by using fractional integrals and derivatives of distributed orders. In order to find a connection between the anomalous diffusion described by these equations and the long-range dependence, we introduced and studied the generalized grey Brownian motion (ggBm), which is actually a parametric class of H-sssi processes, which have indeed marginal probability density function evolving in time according to a partial integro-differential equation of fractional type. The ggBm is of course Non-Markovian. All around the work, we have remarked many times that, starting from a master equation of a probability density function f(x,t), it is always possible to define an equivalence class of stochastic processes with the same marginal density function f(x,t). All these processes provide suitable stochastic models for the starting equation. Studying the ggBm, we just focused on a subclass made up of processes with stationary increments. The ggBm has been defined canonically in the so called grey noise space. However, we have been able to provide a characterization notwithstanding the underline probability space. We also pointed out that that the generalized grey Brownian motion is a direct generalization of a Gaussian process and in particular it generalizes Brownain motion and fractional Brownain motion as well. Finally, we introduced and analyzed a more general class of diffusion type equations related to certain non-Markovian stochastic processes. We started from the forward drift equation, which have been made non-local in time by the introduction of a suitable chosen memory kernel K(t). The resulting non-Markovian equation has been interpreted in a natural way as the evolution equation of the marginal density function of a random time process l(t). We then consider the subordinated process Y(t)=X(l(t)) where X(t) is a Markovian diffusion. The corresponding time-evolution of the marginal density function of Y(t) is governed by a non-Markovian Fokker-Planck equation which involves the same memory kernel K(t). We developed several applications and derived the exact solutions. Moreover, we considered different stochastic models for the given equations, providing path simulations.

Consequences of plant population size for pollinator visitation and plant reproductive success

Habitat loss and fragmentation have a prominent role in determining the size of plant populations, and can affect plant-pollinator interactions. It is hypothesized that in small plant populations the ability to set seeds can be reduced due to limited pollination services, since individuals in small populations can receive less quantity or quality of visits. In this study, I investigated the effect of population size on plant reproductive success and insect visitation in 8 populations of two common species in the island of Lesvos, Greece (Mediterranean Sea), Echium plantagineum and Ballota acetabulosa, and of a rare perennial shrub endemic to north-central Italy, Ononis masquillierii. All the three species depended on insect pollinators for sexual reproduction. For each species, pollen limitation was present in all or nearly all populations, but the relationship between pollen limitation and population size was only present in Ononis masquillierii. However, in Echium plantagineum, significant relationships between both open-pollinated and handcrossed-pollinated seed sets and population size were found, being small populations comparatively less productive than large ones. Additionally, for this species, livestock grazing intensity was greater for small populations and for sparse patches, and had a negative influence on productivity of the remnant plants. Both Echium plantagineum and Ballota acetabulosa attracted a great number of insects, representing a wide spectrum of pollinators, thereby can be considered as generalist species. For Ballota acetabulosa, the most important pollinators were megachilid female bees, and insect diversity didn’t decrease with decreasing plant population size. By contrast, Ononis masquillierii plants generally received few visits, with flowers specialized on small bees (Lasioglossum spp.), representing the most important insect guild. In Echium plantagineum and Ballota acetabulosa, plants in small and large populations received the same amount of visits per flower, and no differences in the number of intraplant visited flowers were detected. On the contrary, large Ononis populations supported higher amounts of pollinators than small ones. At patch level, high Echium flower density was associated with more and higher quality pollinators. My results indicate that small populations were not subject to reduced pollination services than large ones in Echium plantagineum and Ballota acetabulosa, and suggest that grazing and resource limitation could have a major impact on population fitness in Echium plantagineum. The absence of any size effects in these two species can be explained in the light of their high local abundance, wide habitat specificity, and ability to compete with other co-flowering species for pollinators. By contrast, size represents a key characteristic for both pollination and reproduction in Ononis masquillierii populations, as an increase in size could mitigate the negative effects coming from the disadvantageous reproductive traits of the species. Finally, the widespread occurrence of pollen limitation in the three species may be the result of 1) an ongoing weakening or disruption of plantpollinator interactions derived from ecological perturbations, 2) an adaptive equilibrium in response to stochastic processes, and 3) the presence of unfavourable reproductive traits (for Ononis masquillierii).

Sea-Level climate variability in the Mediterranean Sea

Sea-level variability is characterized by multiple interacting factors described in the Fourth Assessment Report (Bindoff et al., 2007) of the Intergovernmental Panel on Climate Change (IPCC) that act over wide spectra of temporal and spatial scales. In Church et al. (2010) sea-level variability and changes are defined as manifestations of climate variability and change. The European Environmental Agency (EEA) defines sea level as one of most important indicators for monitoring climate change, as it integrates the response of different components of the Earths system and is also affected by anthropogenic contributions (EEA, 2011). The balance between the different sea-level contributions represents an important source of uncertainty, involving stochastic processes that are very difficult to describe and understand in detail, to the point that they are defined as an enigma in Munk (2002). Sea-level rate estimates are affected by all these uncertainties, in particular if we look at possible responses to sea-level contributions to future climate. At the regional scale, lateral fluxes also contribute to sea-level variability, adding complexity to sea-level dynamics. The research strategy adopted in this work to approach such an interesting and challenging topic has been to develop an objective methodology to study sea-level variability at different temporal and spatial scales, applicable in each part of the Mediterranean basin in particular, and in the global ocean in general, using all the best calibrated sources of data (for the Mediterranean): in-situ, remote-sensig and numerical models data. The global objective of this work was to achieve a deep understanding of all of the components of the sea-level signal contributing to sea-level variability, tendency and trend and to quantify them.

Model reduction of stochastic groundwater flow and transport processes

This work presents a comprehensive methodology for the reduction of analytical or numerical stochastic models characterized by uncertain input parameters or boundary conditions. The technique, based on the Polynomial Chaos Expansion (PCE) theory, represents a versatile solution to solve direct or inverse problems related to propagation of uncertainty. The potentiality of the methodology is assessed investigating different applicative contexts related to groundwater flow and transport scenarios, such as global sensitivity analysis, risk analysis and model calibration. This is achieved by implementing a numerical code, developed in the MATLAB environment, presented here in its main features and tested with literature examples. The procedure has been conceived under flexibility and efficiency criteria in order to ensure its adaptability to different fields of engineering; it has been applied to different case studies related to flow and transport in porous media. Each application is associated with innovative elements such as (i) new analytical formulations describing motion and displacement of non-Newtonian fluids in porous media, (ii) application of global sensitivity analysis to a high-complexity numerical model inspired by a real case of risk of radionuclide migration in the subsurface environment, and (iii) development of a novel sensitivity-based strategy for parameter calibration and experiment design in laboratory scale tracer transport.

Analysis of ion channel stochastic signals for biosensing

In biological world, life of cells is guaranteed by their ability to sense and to respond to a large variety of internal and external stimuli. In particular, excitable cells, like muscle or nerve cells, produce quick depolarizations in response to electrical, mechanical or chemical stimuli: this means that they can change their internal potential through a quick exchange of ions between cytoplasm and the external environment. This can be done thanks to the presence of ion channels, proteins that span the lipid bilayer and act like switches, allowing ionic current to flow opening and shutting in a stochastic way. For a particular class of ion channels, ligand-gated ion channels, the gating processes is strongly influenced by binding between receptive sites located on the channel surface and specific target molecules. These channels, inserted in biomimetic membranes and in presence of a proper electronic system for acquiring and elaborating the electrical signal, could give us the possibility of detecting and quantifying concentrations of specific molecules in complex mixtures from ionic currents across the membrane; in this thesis work, this possibility is investigated. In particular, it reports a description of experiments focused on the creation and the characterization of artificial lipid membranes, the reconstitution of ion channels and the analysis of their electrical and statistical properties. Moreover, after a chapter about the basis of the modelling of the kinetic behaviour of ligand gated ion channels, a possible approach for the estimation of the target molecule concentration, based on a statistical analysis of the ion channel open probability, is proposed. The fifth chapter contains a description of the kinetic characterisation of a ligand gated ion channel: the homomeric α2 isoform of the glycine receptor. It involved both experimental acquisitions and signal analysis. The last chapter represents the conclusions of this thesis, with some remark on the effective performance that may be achieved using ligand gated ion channels as sensing elements.

A moment symbolic representation of Lévy processes with applications

By using a symbolic method, known in the literature as the classical umbral calculus, a symbolic representation of Lévy processes is given and a new family of time-space harmonic polynomials with respect to such processes, which includes and generalizes the exponential complete Bell polynomials, is introduced. The usefulness of time-space harmonic polynomials with respect to Lévy processes is that it is a martingale the stochastic process obtained by replacing the indeterminate x of the polynomials with a Lévy process, whereas the Lévy process does not necessarily have this property. Therefore to find such polynomials could be particularly meaningful for applications. This new family includes Hermite polynomials, time-space harmonic with respect to Brownian motion, Poisson-Charlier polynomials with respect to Poisson processes, Laguerre and actuarial polynomials with respect to Gamma processes , Meixner polynomials of the first kind with respect to Pascal processes, Euler, Bernoulli, Krawtchuk, and pseudo-Narumi polynomials with respect to suitable random walks. The role played by cumulants is stressed and brought to the light, either in the symbolic representation of Lévy processes and their infinite divisibility property, either in the generalization, via umbral Kailath-Segall formula, of the well-known formulae giving elementary symmetric polynomials in terms of power sum symmetric polynomials. The expression of the family of time-space harmonic polynomials here introduced has some connections with the so-called moment representation of various families of multivariate polynomials. Such moment representation has been studied here for the first time in connection with the time-space harmonic property with respect to suitable symbolic multivariate Lévy processes. In particular, multivariate Hermite polynomials and their properties have been studied in connection with a symbolic version of the multivariate Brownian motion, while multivariate Bernoulli and Euler polynomials are represented as powers of multivariate polynomials which are time-space harmonic with respect to suitable multivariate Lévy processes.